Question

Compare the graphs of the logarithmic functions f left parenthesis x right parenthesis equals log subscript 7 baseline x and g left parenthesis x right parenthesis equals log subscript 4 baseline x. For what values of x is f = g, f greater than g, and f less than g? Explain how you know.
(4 points)

Answer 1

To compare the logarithmic functions \( f(x) = \log_7(x) \) and \( g(x) = \log_4(x) \), we need to analyze their graphs and how they behave over various intervals of \( x \).

### Understanding the Functions

1. **Base of the Logarithm**:
- The function \( f(x) \) is a logarithm with base 7, and \( g(x) \) is a logarithm with base 4. Since 7 is larger than 4, we can expect that for any positive \( x \), \( \log_7(x) \) will grow more slowly than \( \log_4(x) \).

2. **Intercepts**:
- Both functions have an intercept at (1, 0) since \( \log_b(1) = 0 \) for any base \( b \).

### Identifying Points of Intersection

To find the values of \( x \) where \( f(x) = g(x) \):
\[
\log_7(x) = \log_4(x)
\]
Using the change of base formula, we can rewrite this as:
\[
\frac{\log(x)}{\log(7)} = \frac{\log(x)}{\log(4)}
\]
Cross multiplying gives:
\[
\log(x) \cdot \log(4) = \log(x) \cdot \log(7)
\]

If \( \log(x) \neq 0 \) (which is true for \( x \neq 1 \)):
\[
\log(4) = \log(7) \quad \text{(which is false)}
\]
So, the only point where \( f(x) = g(x) \) is at \( x = 1 \).

### Comparing the Functions

Now, we can analyze when \( f(x) \) is greater than or less than \( g(x) \):

1. **For \( x < 1 \)**:
- Both functions are negative because logarithms of numbers less than 1 (but greater than 0) are negative. However, since \( \log(4) > \log(7) \), \( g(x) < f(x) \) in this interval. Thus, for \( 0 < x < 1 \):
\[
f(x) > g(x)
\]

2. **For \( x = 1 \)**:
- As established before, \( f(1) = g(1) = 0 \):
\[
f(x) = g(x)
\]

3. **For \( x > 1 \)**:
- Both functions are positive. The growth rate of \( g(x) = \log_4(x) \) will be greater than that of \( f(x) = \log_7(x) \) due to the smaller base of the logarithm. Therefore, for \( x > 1 \):
\[
f(x) < g(x)
\]

### Summary

- **Values of \( x \) where \( f = g \)**: \( x = 1 \)
- **Values of \( x \) where \( f > g \)**: \( 0 < x < 1 \)
- **Values of \( x \) where \( f < g \)**: \( x > 1 \)

### Conclusion

The graphs of the functions \( f(x) = \log_7(x) \) and \( g(x) = \log_4(x) \) intersect at \( x = 1 \). For values less than 1, \( f \) is greater than \( g \), and for values greater than 1, \( f \) is less than \( g \). This behavior is dictated by the bases of the logarithms: \( \log_4(x) \) outpaces \( \log_7(x) \) as \( x \) increases.